User:Karsten Meyer

I exist. I went to school. I have the "Mittlere Reife" and "Fachhochschulreife". My Profession is Chemical laboratory technician. I went for some Years to the "Fachhochschule Gießen-Friedberg" in Germany. I have written some Articles in "de.wikipedia.org", "de.wikibooks.org" and "en.citizendium.org".

I am interested in Number theory, especially Prime numbers. Pseudoprimes, Lucas sequences and others.

--

Index

Sqequences, where i am participated in:

• OEIS: A111305 - Composite numbers n such that a^(n-1) = 1 mod n only when a = 1 mod n.
• OEIS: A104189 - Prime numbers arising from Schorn's proof that there are infinitely many primes.
• OEIS: A112540 - Numbers n such that (15n-4; 15n-2; 15n+2; 15n+4) is a prime quadruplet.
• OEIS: A051015 - Zeisel numbers.

Comments:

• OEIS: A046388 - Odd numbers with exactly 2 distinct prime factors.
• OEIS: A033181 - Composite n such that abs(a^((n-1)/2) mod n) <> 1 for all a with (a,n) = 1.
• OEIS: A005936 - Pseudoprimes to base 5.
• OEIS: A005938 - Pseudoprimes to base 7.
• OEIS: A020139 - Pseudoprimes to base 11.
• OEIS: A020141 - Pseudoprimes to base 13.
• OEIS: A020145 - Pseudoprimes to base 17.
• OEIS: A020147 - Pseudoprimes to base 19.

1729

1729 is my favorite number because it is not only a Carmichael number. 1729 is the smallest Carmichael number you can construct with the formula $(6n+1)*(12+1)*(18+1)$. 1729 is the smallest absolute euler pseudoprime. 1729 is the product of $7*13$. 1729 is also a Zeisel number.

New Sequence

${\frac {1}{1}}\,{\frac {4}{3}}\,{\frac {9}{6}}\,{\frac {16}{10}}\,{\frac {25}{15}}\,{\frac {36}{21}}\,{\frac {49}{28}}\,{\frac {64}{36}}\,{\frac {81}{45}}\,{\frac {100}{55}}\,{\frac {121}{66}}\,{\frac {144}{78}}\,{\frac {169}{91}}\,{\frac {196}{105}}\,...$

$\sum _{i=1}^{n}{\frac {1}{\Delta i}}={\frac {n^{2}}{\Delta n}}$

$\lim _{n\to \infty }{\frac {n^{2}}{\Delta n}}=2$

Some kinds of Fermat pseudoprimes

Numbers 6n +-1

Fermat Pseudoprimes

Fermat Pseudoprimes sequences (corrected)

Table of Fermat Pseudoprimes

Table of strong Pseudoprimes

Squarefree, odd numbers with exactly 2 distinct prime factors

Super euler pseudoprimes

Carmichael numbers and Zeisel numbers

Lucas Sequences

concat

A143548

User:Karsten Meyer/strong pseudoprimes source

Distribution Primenumbers and fermat Pseudoprimes

Number n	Number of primes p	Number of fermat pseudoprimes q
10	4	0
100	25	27
1000	168	429
10.000	1229	5099

Fingerprint of natural numbers

Fingerprint

Natural numbers

Natural numbers have a common pattern:

	1	2	3	4	5	6
1:	1	1	1	1	1	1
X:	X	X	X	X	X	X
X:	X	X	X	X	X	X
X:	X	X	X	X	X	X
X:	X	X	X	X	X	X
A:	A	1	A	1	A	1

In der obersten Zeile befinden sich immer Einsen und in der untersten Zeile befinden sich immer im Wechsel A und 1, wobei A für $n-1\$ steht. Dies ist also keine Charakteristik, die für Primzahlen typisch ist.

Color	Formular
	$a^{\lambda (n)}\$
	$a^{\frac {\lambda (n)}{2}}$

A prime number has a typical pattern:

	1	2	3	4	5	6
1:	1	1	1	1	1	1
X:	X	X	1	X	X	1
X:	X	X	A	X	X	1
X:	X	X	1	X	X	1
X:	X	X	A	X	X	1
A:	A	1	A	1	A	1

Numbers

1 1: 1	1 2 1: 1 1 2: 2 1	1 2 1: 1 1 2: 2 4 3: 3 1	1 2 3 4 1: 1 1 1 1 2: 2 4 3 1 3: 3 4 2 1 4: 4 1 4 1	1 2 1: 1 1 2: 2 4 3: 3 3 4: 4 4 5: 5 1	1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 1 2 4 1 3: 3 2 6 4 5 1 4: 4 2 1 4 2 1 5: 5 4 6 2 3 1 6: 6 1 6 1 6 1	1 2 1: 1 1 2: 2 4 3: 3 1 4: 4 0 5: 5 1 6: 6 4 7: 7 1	1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 8 7 5 1 3: 3 0 0 0 0 0 4: 4 7 1 4 7 1 5: 5 7 8 4 2 1 6: 6 0 0 0 0 0 7: 7 4 1 7 4 1 8: 8 1 8 1 8 1	1 2 3 4 1: 1 1 1 1 2: 2 4 8 6 3: 3 9 7 1 4: 4 6 4 6 5: 5 5 5 5 6: 6 6 6 6 7: 7 9 3 1 8: 8 4 2 6 9: 9 1 9 1
2	3	4	5	6	7	8	9	10

1 2 3 4 5 6 7 8 9 10 1: 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 5 10 9 7 3 6 1 3: 3 9 5 4 1 3 9 5 4 1 4: 4 5 9 3 1 4 5 9 3 1 5: 5 3 4 9 1 5 3 4 9 1 6: 6 3 7 9 10 5 8 4 2 1 7: 7 5 2 3 10 4 6 9 8 1 8: 8 9 6 4 10 3 2 5 7 1 9: 9 4 3 5 1 9 4 3 5 1 10: 10 1 10 1 10 1 10 1 10 1	1 2 1: 1 1 2: 2 4 3: 3 9 4: 4 4 5: 5 1 6: 6 0 7: 7 1 8: 8 4 9: 9 3 10: 10 4 11: 11 1	1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 3 6 12 11 9 5 10 7 1 3: 3 9 1 3 9 1 3 9 1 3 9 1 4: 4 3 12 9 10 1 4 3 12 9 10 1 5: 5 12 8 1 5 12 8 1 5 12 8 1 6: 6 10 8 9 2 12 7 3 5 4 11 1 7: 7 10 5 9 11 12 6 3 8 4 2 1 8: 8 12 5 1 8 12 5 1 8 12 5 1 9: 9 3 1 9 3 1 9 3 1 9 3 1 10: 10 9 12 3 4 1 10 9 12 3 4 1 11: 11 4 5 3 7 12 2 9 8 10 6 1 12: 12 1 12 1 12 1 12 1 12 1 12 1	1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 8 2 4 8 3: 3 9 13 11 5 1 4: 4 2 8 4 2 8 5: 5 11 13 9 3 1 6: 6 8 6 8 6 8 7: 7 7 7 7 7 7 8: 8 8 8 8 8 8 9: 9 11 1 9 11 1 10: 10 2 6 4 12 8 11: 11 9 1 11 9 1 12: 12 4 6 2 10 8 13: 13 1 13 1 13 1	1 2 3 4 1: 1 1 1 1 2: 2 4 8 1 3: 3 9 2 6 4: 4 1 4 1 5: 5 10 5 10 6: 6 6 6 6 7: 7 4 13 1 8: 8 4 2 1 9: 9 6 9 6 10: 10 10 10 10 11: 11 1 11 1 12: 12 9 3 6 13: 13 4 7 1 14: 14 1 14 1	1 2 3 4 1: 1 1 1 1 2: 2 4 8 0 3: 3 9 11 1 4: 4 0 0 0 5: 5 9 13 1 6: 6 4 8 0 7: 7 1 7 1 8: 8 0 0 0 9: 9 1 9 1 10: 10 4 8 0 11: 11 9 3 1 12: 12 2 8 0 13: 13 9 5 1 14: 14 4 8 0 15: 15 1 15 1
11	12	13	14	15	16

Siebzehn:

Achtzehn:

Neunzehn:

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1:	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	13	7	14	9	18	17	15	11	3	6	12	5	10	1
3:	3	9	8	5	15	7	2	6	18	16	10	11	14	4	12	17	13	1
4:	4	16	7	9	17	11	6	5	1	4	16	7	9	17	11	6	5	1
5:	5	6	11	17	9	7	16	4	1	5	6	11	17	9	7	16	4	1
6:	6	17	7	4	5	11	9	16	1	6	17	7	4	5	11	9	16	1
7:	7	11	1	7	11	1	7	11	1	7	11	1	7	11	1	7	11	1
8:	8	7	18	11	12	1	8	7	18	11	12	1	8	7	18	11	12	1
9:	9	5	7	6	16	11	4	17	1	9	5	7	6	16	11	4	17	1
10:	10	5	12	6	3	11	15	17	18	9	14	7	13	16	8	4	2	1
11:	11	7	1	11	7	1	11	7	1	11	7	1	11	7	1	11	7	1
12:	12	11	18	7	8	1	12	11	18	7	8	1	12	11	18	7	8	1
13:	13	17	12	4	14	11	10	16	18	6	2	7	15	5	8	9	3	1
14:	14	6	8	17	10	7	3	4	18	5	13	11	2	9	12	16	15	1
15:	15	16	12	9	2	11	13	5	18	4	3	7	10	17	8	6	14	1
16:	16	9	11	5	4	7	17	6	1	16	9	11	5	4	7	17	6	1
17:	17	4	11	16	6	7	5	9	1	17	4	11	16	6	7	5	9	1
18:	18	1	18	1	18	1	18	1	18	1	18	1	18	1	18	1	18	1

20:

21:

	1	2	3	4	5	6
1:	1	1	1	1	1	1
2:	2	4	8	16	11	1
3:	3	9	6	18	12	15
4:	4	16	1	4	16	1
5:	5	4	20	16	17	1
6:	6	15	6	15	6	15
7:	7	7	7	7	7	7
8:	8	1	8	1	8	1
9:	9	18	15	9	18	15
10:	10	16	13	4	19	1
11:	11	16	8	4	2	1
12:	12	18	6	9	3	15
13:	13	1	13	1	13	1
14:	14	7	14	7	14	7
15:	15	15	15	15	15	15
16:	16	4	1	16	4	1
17:	17	16	20	4	5	1
18:	18	9	15	18	9	15
19:	19	4	13	16	10	1
20:	20	1	20	1	20	1

Fünfundzwanzig:

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
1:	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	7	14	3	6	12	24	23	21	17	9	18	11	22	19	13	1
3:	3	9	2	6	18	4	12	11	8	24	22	16	23	19	7	21	13	14	17	1
4:	4	16	14	6	24	21	9	11	19	1	4	16	14	6	24	21	9	11	19	1
5:	5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
6:	6	11	16	21	1	6	11	16	21	1	6	11	16	21	1	6	11	16	21	1
7:	7	24
8:	8	14
9:	9	6	4	11
10:	10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
11:	11	21
12:	12	1	20	1	20	1
13:	13	1	1	1	1	1
14:	14	4	8	16	11	1
15:	15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
16:	16	4	20	16	17	1
17:	17	1	8	1	8	1
18:	18	16	13	4	19	1
19:	19	16	8	4	2	1
20:	20	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
21:	21	4	1	16	4	1
22:	22	16	20	4	5	1
23:	23	4	13	16	10	1
24:	24	1	24	1	24	1	24	1	24	1	24	1	24	1	24	1	24	1	24	1

Achtundzwanzig:

	1	2	3	4	5	6
1:	1	1	1	1	1	1
2:	2	4	8	16	4	8
3:	3	9	27	25	19	1
4:	4	16	8	4	16	8
5:	5	25	13	9	17	1
6:	6	8	20	8	20	8
7:	7	21	7	21	7	21
8:	8	8	8	8	8	8
9:	9	25	1	9	25	1
10:	10	16	20	4	12	8
11:	11	9	15	25	23	1
12:	12	4	20	16	24	8
13:	13	1	13	1	13	1
14:	14	0	0	0	0	0
15:	15	1	15	1	15	1
16:	16	4	8	16	4	8
17:	17	9	13	25	5	1
18:	18	16	8	4	16	8
19:	19	25	27	9	3	1
20:	20	8	20	8	20	8
21:	21	21	21	21	21	21
22:	22	8	8	8	8	8
23:	23	25	15	9	11	1
24:	24	16	20	4	12	8
25:	25	9	1	25	9	1
26:	26	4	20	16	24	8
27:	27	1	27	1	27	1

33:

	1	2	3	4	5	6	7	8	9	10
1:	1	1	1	1	1	1	1	1	1	1 1
2:	2	4	8	16	32	31	29	25	17	1
3:	3	9	27	15	12	3	9	27	15	12
4:	4	16	31	25	1	4	16	31	25	1
5:	5	25	26	31	23	16	14	4	20	1
6:	6	3	18	9	21	27	30	15	24	12
7:	7	16	13	25	10	4	28	31	19	1
8:	8	31	17	4	32	25	2	16	29	1
9:	9	15	3	27	12	9	15	3	27	12
10:	10	1	10	1	10	1	10	1	10	1
11:	11	22	11	22	11	22	11	22	11	22
12:	12	12	12	12	12	12	12	12	12	12
13:	13	4	19	16	10	31	7	25	28	1
14:	14	31	5	4	23	25	20	16	26	1
15:	15	27	9	3	12	15	27	9	3	12
16:	16	25	4	31	1	16	25	4	31	1
17:	17	25	29	31	32	16	8	4	2	1
18:	18	27	24	3	21	15	6	9	30	12
19:	19	31	28	4	10	25	13	16	7	1
20:	20	4	14	16	23	31	26	25	5	1
21:	21	12	21	12	21	12	21	12	21	12
22:	22	22	22	22	22	22	22	22	22	22
23:	23	1	23	1	23	1	23	1	23	1
24:	24	15	30	27	21	9	18	3	6	12
25:	25	31	16	4	1	25	31	16	4	1
26:	26	16	20	25	23	4	5	31	14	1
27:	27	3	15	9	12	27	3	15	9	12
28:	28	25	7	31	10	16	19	4	13	1
29:	29	16	2	25	32	4	17	31	8	1
30:	30	9	6	15	21	3	24	27	18	12
31:	31	4	25	16	1	31	4	25	16	1
32:	32	1	32	1	32	1	32	1	32	1 1

35

	1	2	3	4	5	6	7	8	9	10	11	12
1:	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	29	23	11	22	9	18	1
3:	3	9	27	11	33	29	17	16	13	4	12	1
4:	4	16	29	11	9	1	4	16	29	11	9	1
5:	5	25	20	30	10	15	5	25	20	30	10	15
6:	6	1	6	1	6	1	6	1	6	1	6	1
7:	7	14	28	21	7	14	28	21	7	14	28	21
8:	8	29	22	1	8	29	22	1	8	29	22	1
9:	9	11	29	16	4	1	9	11	29	16	4	1
10:	10	30	20	25	5	15	10	30	20	25	5	15
11:	11	16	1	11	16	1	11	16	1	11	16	1
12:	12	4	13	16	17	29	33	11	27	9	3	1
13:	13	29	27	1	13	29	27	1	13	29	27	1
14:	14	21	14	21	14	21	14	21	14	21	14	21
15:	15	15	15	15	15	15	15	15	15	15	15	15
16:	16	11	1	16	11	1	16	11	1	16	11	1
17:	17	9	13	11	12	29	3	16	27	4	33	1
18:	18	9	22	11	23	29	32	16	8	4	2	1
19:	19	11	34	16	24	1	19	11	34	16	24	1
20:	20	15	20	15	20	15	20	15	20	15	20	15
21:	21	21	21	21	21	21	21	21	21	21	21	21
22:	22	29	8	1	22	29	8	1	22	29	8	1
23:	23	4	22	16	18	29	2	11	8	9	32	1
24:	24	16	34	11	19	1	24	16	34	11	19	1
25:	25	30	15	25	30	15	25	30	15	25	30	15
26:	26	11	6	16	31	1	26	11	6	16	31	1
27:	27	29	13	1	27	29	13	1	27	29	13	1
28:	28	14	7	21	28	14	7	21	28	14	7	21
29:	29	1	29	1	29	1	29	1	29	1	29	1
30:	30	25	15	30	25	15	30	25	15	30	25	15
31:	31	16	6	11	26	1	31	16	6	11	26	1
32:	32	9	8	11	2	29	18	16	22	4	23	1
33:	33	4	27	16	3	29	12	11	13	9	17	1
34:	34	1	34	1	34	1	34	1	34	1	34	1

39

	1	2	3	4	5	6	7	8	9	10	11	12
1:	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	25	11	22	5	10	20	1
3:	3	9	27	3	9	27	3	9	27	3	9	27
4:	4	16	25	22	10	1	4	16	25	22	10	1
5:	5	25	8	1	5	25	8	1	5	25	8	1
6:	6	36	21	9	15	12	33	3	18	30	24	27
7:	7	10	31	22	37	25	19	16	34	4	28	1
8:	8	25	5	1	8	25	5	1	8	25	5	1
9:	9	3	27	9	3	27	9	3	27	9	3	27
10:	10	22	25	16	4	1	10	22	25	16	4	1
11:	11	4	5	16	20	25	2	22	8	10	32	1
12:	12	27	12	27	12	27	12	27	12	27	12	27
13:	13	13	13	13	13	13	13	13	13	13	13	13
14:	14	1	14	1	14	1	14	1	14	1	14	1
15:	15	30	21	3	6	12	24	9	18	36	33	27
16:	16	22	1	16	22	1	16	22	1	16	22	1
17:	17	16	38	22	23	1	17	16	38	22	23	1
18:	18	12	21	27	18	12	21	27	18	12	21	27
19:	19	10	34	22	28	25	7	16	31	4	37	1
20:	20	10	5	22	11	25	32	16	8	4	2	1
21:	21	12	18	27	21	12	18	27	21	12	18	27
22:	22	16	1	22	16	1	22	16	1	22	16	1
23:	23	22	38	16	17	1	23	22	38	16	17	1
24:	24	30	18	3	33	12	15	9	21	36	6	27
25:	25	1	25	1	25	1	25	1	25	1	25	1
26:	26	13	26	13	26	13	26	13	26	13	26	13
27:	27	27	27	27	27	27	27	27	27	27	27	27
28:	28	4	34	16	19	25	37	22	31	10	7	1
29:	29	22	14	16	35	1	29	22	14	16	35	1
30:	30	3	12	9	36	27	30	3	12	9	36	27
31:	31	25	34	1	31	25	34	1	31	25	34	1
32:	32	10	8	22	2	25	20	16	5	4	11	1
33:	33	36	18	9	24	12	6	3	21	30	15	27
34:	34	25	31	1	34	25	31	1	34	25	31	1
35:	35	16	14	22	29	1	35	16	14	22	29	1
36:	36	9	12	3	30	27	36	9	12	3	30	27
37:	37	4	31	16	7	25	28	22	34	10	19	1
38:	38	1	38	1	38	1	38	1	38	1	38	1

45

	1	2	3	4	5	6	7	8	9	10	11	12
1:	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	19	38	31	17	34	23	1
3:	3	9	27	36	18	9	27	36	18	9	27	36
4:	4	16	19	31	34	1	4	16	19	31	34	1
5:	5	25	35	40	20	10	5	25	35	40	20	10
6:	6	36	36	36	36	36	36	36	36	36	36	36
7:	7	4	28	16	22	19	43	31	37	34	13	1
8:	8	19	17	1	8	19	17	1	8	19	17	1
9:	9	36	9	36	9	36	9	36	9	36	9	36
10:	10	10	10	10	10	10	10	10	10	10	10	10
11:	11	31	26	16	41	1	11	31	26	16	41	1
12:	12	9	18	36	27	9	18	36	27	9	18	36
13:	13	34	37	31	43	19	22	16	28	4	7	1
14:	14	16	44	31	29	1	14	16	44	31	29	1
15:	15	0	0	0	0	0	0	0	0	0	0	0
16:	16	31	1	16	31	1	16	31	1	16	31	1
17:	17	19	8	1	17	19	8	1	17	19	8	1
18:	18	9	27	36	18	9	27	36	18	9	27	36
19:	19	1	19	1	19	1	19	1	19	1	19	1
20:	20	40	35	25	5	10	20	40	35	25	5	10
21:	21	36	36	36	36	36	36	36	36	36	36	36
22:	22	34	28	31	7	19	13	16	37	4	43	1
23:	23	34	17	31	38	19	32	16	8	4	2	1
24:	24	36	9	36	9	36	9	36	9	36	9	36
25:	25	40	10	25	40	10	25	40	10	25	40	10
26:	26	1	26	1	26	1	26	1	26	1	26	1
27:	27	9	18	36	27	9	18	36	27	9	18	36
28:	28	19	37	1	28	19	37	1	28	19	37	1
29:	29	31	44	16	14	1	29	31	44	16	14	1
30:	30	0	0	0	0	0	0	0	0	0	0	0
31:	31	16	1	31	16	1	31	16	1	31	16	1
32:	32	34	8	31	2	19	23	16	17	4	38	1
33:	33	9	27	36	18	9	27	36	18	9	27	36
34:	34	31	19	16	4	1	34	31	19	16	4	1
35:	35	10	35	10	35	10	35	10	35	10	35	10
36:	36	36	36	36	36	36	36	36	36	36	36	36
37:	37	19	28	1	37	19	28	1	37	19	28	1
38:	38	4	17	16	23	19	2	31	8	34	32	1
39:	39	36	9	36	9	36	9	36	9	36	9	36
40:	40	25	10	40	25	10	40	25	10	40	25	10
41:	41	16	26	31	11	1	41	16	26	31	11	1
42:	42	9	18	36	27	9	18	36	27	9	18	36
43:	43	4	37	16	13	19	7	31	28	34	22	1
44:	44	1	44	1	44	1	44	1	44	1	44	1

49

51

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1:	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	13	26	1	2	4	8	16	32	13	26	1
3:	3	9	27	30	39	15	45	33	48	42	24	21	12	36	6	18
4:	4	16	13	1	4	16	13	1	4	16	13	1	4	16	13	1
5:	5	25	23	13	14	19	44	16	29	43	11	4	20	49	41	1
6:	6	36	12	21	24	42	48	33	45	15	39	30	27	9	3	18
7:	7	49	37	4	28	43	46	16	10	19	31	13	40	25	22	1
8:	8	13	2	16	26	4	32	1	8	13	2	16	26	4	32	1
9:	9	30	15	33	42	21	36	18	9	30	15	33	42	21	36	18
10:	10	49	31	4	40	43	22	16	7	19	37	13	28	25	46	1
11:	11	19	5	4	44	25	20	16	23	49	29	13	41	43	14	1
12:	12	42	45	30	3	36	24	33	39	9	6	21	48	15	27	18
13:	13	16	4	1	13	16	4	1	13	16	4	1	13	16	4	1
14:	14	43	41	13	29	49	23	16	20	25	44	4	5	19	11	1
15:	15	21	9	33	36	30	42	18	15	21	9	33	36	30	42	18
16:	16	1	16	1	16	1	16	1	16	1	16	1	16	1	16	1
17:	17	34	17	34	17	34	17	34	17	34	17	34	17	34	17	34
18:	18	18	18	18	18	18	18	18	18	18	18	18	18	18	18	18
19:	19	4	25	16	49	13	43	1	19	4	25	16	49	13	43	1
20:	20	43	44	13	5	49	11	16	14	25	41	4	29	19	23	1
21:	21	33	30	18	21	33	30	18	21	33	30	18	21	33	30	18
22:	22	25	40	13	31	19	10	16	46	43	28	4	37	49	7	1
23:	23	19	29	4	41	25	14	16	11	49	5	13	44	43	20	1
24:	24	15	3	21	45	9	12	33	27	36	48	30	6	42	39	18
25:	25	13	19	16	43	4	49	1	25	13	19	16	43	4	49	1
26:	26	13	32	16	8	4	2	1	26	13	32	16	8	4	2	1
27:	27	15	48	21	6	9	39	33	24	36	3	30	45	42	12	18
28:	28	19	22	4	10	25	37	16	40	49	46	13	7	43	31	1
29:	29	25	11	13	20	19	41	16	5	43	23	4	14	49	44	1
30:	30	33	21	18	30	33	21	18	30	33	21	18	30	33	21	18
31:	31	43	7	13	46	49	40	16	37	25	10	4	22	19	28	1
32:	32	4	26	16	2	13	8	1	32	4	26	16	2	13	8	1
33:	33	18	33	18	33	18	33	18	33	18	33	18	33	18	33	18
34:	34	34	34	34	34	34	34	34	34	34	34	34	34	34	34	34
35:	35	1	35	1	35	1	35	1	35	1	35	1	35	1	35	1
36:	36	21	42	33	15	30	9	18	36	21	42	33	15	30	9	18
37:	37	43	10	13	22	49	28	16	31	25	7	4	46	19	40	1
38:	38	16	47	1	38	16	47	1	38	16	47	1	38	16	47	1
39:	39	42	6	30	48	36	27	33	12	9	45	21	3	15	24	18
40:	40	19	46	4	7	25	31	16	28	49	22	13	10	43	37	1
41:	41	49	20	4	11	43	29	16	44	19	14	13	23	25	5	1
42:	42	30	36	33	9	21	15	18	42	30	36	33	9	21	15	18
43:	43	13	49	16	25	4	19	1	43	13	49	16	25	4	19	1
44:	44	49	14	4	23	43	5	16	41	19	20	13	11	25	29	1
45:	45	36	39	21	27	42	3	33	6	15	12	30	24	9	48	18
46:	46	25	28	13	37	19	7	16	22	43	40	4	31	49	10	1
47:	47	16	38	1	47	16	38	1	47	16	38	1	47	16	38	1
48:	48	9	24	30	12	15	6	33	3	42	27	21	39	36	45	18
49:	49	4	43	16	19	13	25	1	49	4	43	16	19	13	25	1
50:	50	1	50	1	50	1	50	1	50	1	50	1	50	1	50	1

52

	1	2	3	4	5	6	7	8	9	10	11	12
1:	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	12	24	48	44	36	20	40
3:	3	9	27	29	35	1	3	9	27	29	35	1
4:	4	16	12	48	36	40	4	16	12	48	36	40
5:	5	25	21	1	5	25	21	1	5	25	21	1
6:	6	36	8	48	28	12	20	16	44	4	24	40
7:	7	49	31	9	11	25	19	29	47	17	15	1
8:	8	12	44	40	8	12	44	40	8	12	44	40
9:	9	29	1	9	29	1	9	29	1	9	29	1
10:	10	48	12	16	4	40	36	48	12	16	4	40
11:	11	17	31	29	7	25	15	9	47	49	19	1
12:	12	40	12	40	12	40	12	40	12	40	12	40
13:	13	13	13	13	13	13	13	13	13	13	13	13
14:	14	40	40	40	40	40	40	40	40	40	40	40
15:	15	17	47	29	19	25	11	9	31	49	7	1
16:	16	48	40	16	48	40	16	48	40	16	48	40
17:	17	29	25	9	49	1	17	29	25	9	49	1
18:	18	12	8	40	44	12	8	40	44	12	8	40
19:	19	49	47	9	15	25	7	29	31	17	11	1
20:	20	36	44	48	24	12	32	16	8	4	28	40
21:	21	25	5	1	21	25	5	1	21	25	5	1
22:	22	16	40	48	16	40	48	16	40	48	16	40
23:	23	9	51	29	43	1	23	9	51	29	43	1
24:	24	4	44	16	20	12	28	48	8	36	32	40
25:	25	1	25	1	25	1	25	1	25	1	25	1
26:	26	.0	.0	.0	.0	.0	.0	.0	.0	.0	.0	.0
27:	27	1	27	1	27	1	27	1	27	1	27	1
28:	28	4	8	16	32	12	24	48	44	36	20	40
29:	29	9	1	29	9	1	29	9	1	29	9	1
30:	30	16	12	48	36	40	4	16	12	48	36	40
31:	31	25	47	1	31	25	47	1	31	25	47	1
32:	32	36	8	48	28	12	20	16	44	4	24	40
33:	33	49	5	9	37	25	45	29	21	17	41	1
34:	34	12	44	40	8	12	44	40	8	12	44	40
35:	35	29	27	9	3	1	35	29	27	9	3	1
36:	36	48	12	16	4	40	36	48	12	16	4	40
37:	37	17	5	29	33	25	41	9	21	49	45	1
38:	38	40	12	40	12	40	12	40	12	40	12	40
39:	39	13	39	13	39	13	39	13	39	13	39	13
40:	40	40	40	40	40	40	40	40	40	40	40	40
41:	41	17	21	29	45	25	37	9	5	49	33	1
42:	42	48	40	16	48	40	16	48	40	16	48	40
43:	43	29	51	9	23	1	43	29	51	9	23	1
44:	44	12	8	40	44	12	8	40	44	12	8	40
45:	45	49	21	9	41	25	33	29	5	17	37	1
46:	46	36	44	48	24	12	32	16	8	4	28	40
47:	47	25	31	1	47	25	31	1	47	25	31	1
48:	48	16	40	48	16	40	48	16	40	48	16	40
49:	49	9	25	29	17	1	49	9	25	29	17	1
50:	50	4	44	16	20	12	28	48	8	36	32	40
51:	51	1	51	1	51	1	51	1	51	1	51	1

55

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
1:	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	9	18	36	17	34	13	26	52	49	43	31	7	14	28	1
3:	3	9	27	26	23	14	42	16	48	34	47	31	38	4	12	36	53	49	37	1
4:	4	16	9	36	34	26	49	31	14	1	4	16	9	36	34	26	49	31	14	1
5:	5	25	15	20	45	5	25	15	20	45	5	25	15	20	45	5	25	15	20	45
6:	6	36	51	31	21	16	41	26	46	1	6	36	51	31	21	16	41	26	46	1
7:	7	49	13	36	32	4	28	31	52	34	18	16	2	14	43	26	17	9	8	1
8:	8	9	17	26	43	14	2	16	18	34	52	31	28	4	32	36	13	49	7	1
9:	9	26	14	16	34	31	4	36	49	1	9	26	14	16	34	31	4	36	49	1
10:	10	45	10	45	10	45	10	45	10	45	10	45	10	45	10	45	10	45	10	45
11:	11	11	11	11	11	11	11	11	11	11	11	11	11	11	11	11	11	11	11	11
12:	12	34	23	1	12	34	23	1	12	34	23	1	12	34	23	1	12	34	23	1
13:	13	4	52	16	43	9	7	36	28	34	2	26	8	49	32	31	18	14	17	1
14:	14	31	49	26	34	36	9	16	4	1	14	31	49	26	34	36	9	16	4	1
15:	15	5	20	25	45	15	5	20	25	45	15	5	20	25	45	15	5	20	25	45
16:	16	36	26	31	1	16	36	26	31	1	16	36	26	31	1	16	36	26	31	1
17:	17	14	18	31	32	49	8	26	2	34	28	36	7	9	43	16	52	4	13	1
18:	18	49	2	36	43	4	17	31	8	34	7	16	13	14	32	26	28	9	52	1
19:	19	31	39	26	54	36	24	16	29	1	19	31	39	26	54	36	24	16	29	1
20:	20	15	25	5	45	20	15	25	5	45	20	15	25	5	45	20	15	25	5	45
21:	21	1	21	1	21	1	21	1	21	1	21	1	21	1	21	1	21	1	21	1
22:	22	44	33	11	22	44	33	11	22	44	33	11	22	44	33	11	22	44	33	11
23:	23	34	12	1	23	34	12	1	23	34	12	1	23	34	12	1	23	34	12	1
24:	24	26	19	16	54	31	29	36	39	1	24	26	19	16	54	31	29	36	39	1
25:	25	20	5	15	45	25	20	5	15	45	25	20	5	15	45	25	20	5	15	45
26:	26	16	31	36	1	26	16	31	36	1	26	16	31	36	1	26	16	31	36	1
27:	27	14	48	31	12	49	3	26	42	34	38	36	37	9	23	16	47	4	53	1
28:	28	14	7	31	43	49	52	26	13	34	17	36	18	9	32	16	8	4	2	1
29:	29	16	24	36	54	26	39	31	19	1	29	16	24	36	54	26	39	31	19	1
30:	30	20	50	15	10	25	35	5	40	45	30	20	50	15	10	25	35	5	40	45
31:	31	26	36	16	1	31	26	36	16	1	31	26	36	16	1	31	26	36	16	1
32:	32	34	43	1	32	34	43	1	32	34	43	1	32	34	43	1	32	34	43	1
33:	33	44	22	11	33	44	22	11	33	44	22	11	33	44	22	11	33	44	22	11
34:	34	1	34	1	34	1	34	1	34	1	34	1	34	1	34	1	34	1	34	1
35:	35	15	30	5	10	20	40	25	50	45	35	15	30	5	10	20	40	25	50	45
36:	36	31	16	26	1	36	31	16	26	1	36	31	16	26	1	36	31	16	26	1
37:	37	49	53	36	12	4	38	31	47	34	48	16	42	14	23	26	27	9	3	1
38:	38	14	37	31	23	49	47	26	53	34	27	36	48	9	12	16	3	4	42	1
39:	39	36	29	31	54	16	19	26	24	1	39	36	29	31	54	16	19	26	24	1
40:	40	5	35	25	10	15	50	20	30	45	40	5	35	25	10	15	50	20	30	45
41:	41	31	6	26	21	36	46	16	51	1	41	31	6	26	21	36	46	16	51	1
42:	42	4	3	16	12	9	48	36	27	34	53	26	47	49	23	31	37	14	38	1
43:	43	34	32	1	43	34	32	1	43	34	32	1	43	34	32	1	43	34	32	1
44:	44	11	44	11	44	11	44	11	44	11	44	11	44	11	44	11	44	11	44	11
45:	45	45	45	45	45	45	45	45	45	45	45	45	45	45	45	45	45	45	45	45
46:	46	26	41	16	21	31	51	36	6	1	46	26	41	16	21	31	51	36	6	1
47:	47	9	38	26	12	14	53	16	37	34	3	31	27	4	23	36	42	49	48	1
48:	48	49	42	36	23	4	27	31	3	34	37	16	53	14	12	26	38	9	47	1
49:	49	36	4	31	34	16	14	26	9	1	49	36	4	31	34	16	14	26	9	1
50:	50	25	40	20	10	5	30	15	35	45	50	25	40	20	10	5	30	15	35	45
51:	51	16	46	36	21	26	6	31	41	1	51	16	46	36	21	26	6	31	41	1
52:	52	9	28	26	32	14	13	16	7	34	8	31	17	4	43	36	2	49	18	1
53:	53	4	47	16	23	9	37	36	38	34	42	26	3	49	12	31	48	14	27	1
54:	54	1	54	1	54	1	54	1	54	1	54	1	54	1	54	1	54	1	54	1

57

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1:	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	7	14	28	56	55	53	49	41	25	50	43	29	1
3:	3	9	27	24	15	45	21	6	18	54	48	30	33	42	12	36	51	39
4:	4	16	7	28	55	49	25	43	1	4	16	7	28	55	49	25	43	1
5:	5	25	11	55	47	7	35	4	20	43	44	49	17	28	26	16	23	1
6:	6	36	45	42	24	30	9	54	39	6	36	45	42	24	30	9	54	39
7:	7	49	1	7	49	1	7	49	1	7	49	1	7	49	1	7	49	1
8:	8	7	56	49	50	1	8	7	56	49	50	1	8	7	56	49	50	1
9:	9	24	45	6	54	30	42	36	39	9	24	45	6	54	30	42	36	39
10:	10	43	31	25	22	49	34	55	37	28	52	7	13	16	46	4	40	1
11:	11	7	20	49	26	1	11	7	20	49	26	1	11	7	20	49	26	1
12:	12	30	18	45	27	39	12	30	18	45	27	39	12	30	18	45	27	39
13:	13	55	31	4	52	49	10	16	37	25	40	7	34	43	46	28	22	1
14:	14	25	8	55	29	7	41	4	56	43	32	49	2	28	50	16	53	1
15:	15	54	12	9	21	30	51	24	18	42	3	45	48	36	27	6	33	39
16:	16	28	49	43	4	7	55	25	1	16	28	49	43	4	7	55	25	1
17:	17	4	11	16	44	7	5	28	20	55	23	49	35	25	26	43	47	1
18:	18	39	18	39	18	39	18	39	18	39	18	39	18	39	18	39	18	39
19:	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19
20:	20	1	20	1	20	1	20	1	20	1	20	1	20	1	20	1	20	1
21:	21	42	27	54	51	45	33	9	18	36	15	30	3	6	12	24	48	39
22:	22	28	46	43	34	7	40	25	37	16	10	49	52	4	31	55	13	1
23:	23	16	26	28	17	49	44	43	20	4	35	7	47	55	11	25	5	1
24:	24	6	30	36	9	45	54	42	39	24	6	30	36	9	45	54	42	39
25:	25	55	7	4	43	49	28	16	1	25	55	7	4	43	49	28	16	1
26:	26	49	20	7	11	1	26	49	20	7	11	1	26	49	20	7	11	1
27:	27	45	18	30	12	39	27	45	18	30	12	39	27	45	18	30	12	39
28:	28	43	7	25	16	49	4	55	1	28	43	7	25	16	49	4	55	1
29:	29	43	50	25	41	49	53	55	56	28	14	7	32	16	8	4	2	1
30:	30	45	39	30	45	39	30	45	39	30	45	39	30	45	39	30	45	39
31:	31	49	37	7	46	1	31	49	37	7	46	1	31	49	37	7	46	1
32:	32	55	50	4	14	49	29	16	56	25	2	7	53	43	8	28	41	1
33:	33	6	27	36	48	45	3	42	18	24	51	30	21	9	12	54	15	39
34:	34	16	31	28	40	49	13	43	37	4	22	7	10	55	46	25	52	1
35:	35	28	11	43	23	7	17	25	20	16	47	49	5	4	26	55	44	1
36:	36	42	30	54	6	45	24	9	39	36	42	30	54	6	45	24	9	39
37:	37	1	37	1	37	1	37	1	37	1	37	1	37	1	37	1	37	1
38:	38	19	38	19	38	19	38	19	38	19	38	19	38	19	38	19	38	19
39:	39	39	39	39	39	39	39	39	39	39	39	39	39	39	39	39	39	39
40:	40	4	46	16	13	7	52	28	37	55	34	49	22	25	31	43	10	1
41:	41	28	8	43	53	7	2	25	56	16	29	49	14	4	50	55	32	1
42:	42	54	45	9	36	30	6	24	39	42	54	45	9	36	30	6	24	39
43:	43	25	49	55	28	7	16	4	1	43	25	49	55	28	7	16	4	1
44:	44	55	26	4	5	49	47	16	20	25	17	7	23	43	11	28	35	1
45:	45	30	39	45	30	39	45	30	39	45	30	39	45	30	39	45	30	39
46:	46	7	37	49	31	1	46	7	37	49	31	1	46	7	37	49	31	1
47:	47	43	26	25	35	49	23	55	20	28	5	7	44	16	11	4	17	1
48:	48	24	12	6	3	30	15	36	18	9	33	45	51	54	27	42	21	39
49:	49	7	1	49	7	1	49	7	1	49	7	1	49	7	1	49	7	1
50:	50	49	56	7	8	1	50	49	56	7	8	1	50	49	56	7	8	1
51:	51	36	12	42	33	30	48	54	18	6	21	45	15	24	27	9	3	39
52:	52	25	46	55	10	7	22	4	37	43	13	49	40	28	31	16	34	1
53:	53	16	50	28	2	49	32	43	56	4	41	7	29	55	8	25	14	1
54:	54	9	30	24	42	45	36	6	39	54	9	30	24	42	45	36	6	39
55:	55	4	49	16	25	7	43	28	1	55	4	49	16	25	7	43	28	1
56:	56	1	56	1	56	1	56	1	56	1	56	1	56	1	56	1	56	1

65:

	1	2	3	4	5	6	7	8	9	10	11	12
1:	1	1	1	1	1	1	1	1	1	1	1	1
2:	2	4	8	16	32	64	63	61	57	49	33	1
3:	3	9	27	16	48	14	42	61	53	29	22	1
4:	4	16	64	61	49	1	4	16	64	61	49	1
5:	5	25	60	40	5	25	60	40	5	25	60	40
6:	6	36	21	61	41	51	46	16	31	56	11	1
7:	7	49	18	61	37	64	58	16	47	4	28	1
8:	8	64	57	1	8	64	57	1	8	64	57	1
9:	9	16	14	61	29	1	9	16	14	61	29	1
10:	10	35	25	55	30	40	10	35	25	55	30	40
11:	11	56	31	16	46	51	41	61	21	36	6	1
12:	12	14	38	1	12	14	38	1	12	14	38	1
13:	13	39	52	26	13	39	52	26	13	39	52	26
14:	14	1	14	1	14	1	14	1	14	1	14	1
15:	15	30	60	55	45	25	50	35	5	10	20	40
16:	16	61	1	16	61	1	16	61	1	16	61	1
17:	17	29	38	61	62	14	43	16	12	9	23	1
18:	18	64	47	1	18	64	47	1	18	64	47	1
19:	19	36	34	61	54	51	59	16	44	56	24	1
20:	20	10	5	35	50	25	45	55	60	30	15	40
21:	21	51	31	1	21	51	31	1	21	51	31	1
22:	22	29	53	61	42	14	48	16	27	9	3	1
23:	23	9	12	16	43	14	62	61	38	29	17	1
24:	24	56	44	16	59	51	54	61	34	36	19	1
25:	25	40	25	40	25	40	25	40	25	40	25	40
26:	26	26	26	26	26	26	26	26	26	26	26	26
27:	27	14	53	1	27	14	53	1	27	14	53	1
28:	28	4	47	16	58	64	37	61	18	49	7	1
29:	29	61	14	16	9	1	29	61	14	16	9	1
30:	30	55	25	35	10	40	30	55	25	35	10	40
31:	31	51	21	1	31	51	21	1	31	51	21	1
32:	32	49	8	61	2	64	33	16	57	4	63	1
33:	33	49	57	61	63	64	32	16	8	4	2	1
34:	34	51	44	1	34	51	44	1	34	51	44	1
35:	35	55	40	35	55	40	35	55	40	35	55	40
36:	36	61	51	16	56	1	36	61	51	16	56	1
37:	37	4	18	16	7	64	28	61	47	49	58	1
38:	38	14	12	1	38	14	12	1	38	14	12	1
39:	39	26	39	26	39	26	39	26	39	26	39	26
40:	40	40	40	40	40	40	40	40	40	40	40	40
41:	41	56	21	16	6	51	11	61	31	36	46	1
42:	42	9	53	16	22	14	3	61	27	29	48	1
43:	43	29	12	61	23	14	17	16	38	9	62	1
44:	44	51	34	1	44	51	34	1	44	51	34	1
45:	45	10	60	35	15	25	20	55	5	30	50	40
46:	46	36	31	61	11	51	6	16	21	56	41	1
47:	47	64	18	1	47	64	18	1	47	64	18	1
48:	48	29	27	61	3	14	22	16	53	9	42	1
49:	49	61	64	16	4	1	49	61	64	16	4	1
50:	50	30	5	55	20	25	15	35	60	10	45	40
51:	51	1	51	1	51	1	51	1	51	1	51	1
52:	52	39	13	26	52	39	13	26	52	39	13	26
53:	53	14	27	1	53	14	27	1	53	14	27	1
54:	54	56	34	16	19	51	24	61	44	36	59	1
55:	55	35	40	55	35	40	55	35	40	55	35	40
56:	56	16	51	61	36	1	56	16	51	61	36	1
57:	57	64	8	1	57	64	8	1	57	64	8	1
58:	58	49	47	61	28	64	7	16	18	4	37	1
59:	59	36	44	61	24	51	19	16	34	56	54	1
60:	60	25	5	40	60	25	5	40	60	25	5	40
61:	61	16	1	61	16	1	61	16	1	61	16	1
62:	62	9	38	16	17	14	23	61	12	29	43	1
63:	63	4	57	16	33	64	2	61	8	49	32	1
64:	64	1	64	1	64	1	64	1	64	1	64	1